Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I

AbstractUnder the local Langlands correspondence, the conductor of an irreducible representation of Gln(F) is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

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A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2009/749631 ◽

2009 ◽

Vol 2009 ◽

pp. 1-14

Author(s):

Do Ngoc Diep

Keyword(s):

Symmetry Group ◽

Lie Algebra ◽

Vector Bundles ◽

Sigma Model ◽

Fock Space ◽

Loop Algebra ◽

Target Space ◽

Automorphic Representations ◽

Langlands Correspondence ◽

Geometric Langlands

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry groupGL. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundles with connection over the space dual to the Lie algebra of the symmetry groupGL. After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry groupGL. Use the electric-magnetic duality to pass to the Langlands dual Lie groupG. Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra=Lie(G). Use the construction of Fock space reprsentations to have representations of such affine loop algebra. And finally, we have the automorphic representations of the corresponding Langlands-dual Lie groupsG.

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Introduction

Berkeley Lectures on p-adic Geometry ◽

10.23943/princeton/9780691202082.003.0001 ◽

2020 ◽

pp. 1-6

Author(s):

Peter Scholze ◽

Jared Weinstein

Keyword(s):

Moduli Spaces ◽

Function Fields ◽

Number Fields ◽

Important Special Case ◽

Langlands Correspondence ◽

Key Innovation ◽

Perfectoid Spaces ◽

Definition Of ◽

Special Case

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

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On a vanishing conjecture appearing in the geometric Langlands correspondence

Annals of Mathematics ◽

10.4007/annals.2004.160.617 ◽

2004 ◽

Vol 160 (2) ◽

pp. 617-682 ◽

Cited By ~ 8

Author(s):

Dennis Gaitsgory

Keyword(s):

Langlands Correspondence ◽

Geometric Langlands

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LIOUVILLE THEORY: WARD IDENTITIES FOR GENERATING FUNCTIONAL AND MODULAR GEOMETRY

Modern Physics Letters A ◽

10.1142/s021773239400215x ◽

1994 ◽

Vol 09 (25) ◽

pp. 2293-2299 ◽

Cited By ~ 10

Author(s):

LEON A. TAKHTAJAN

Keyword(s):

Moduli Spaces ◽

Correlation Functions ◽

Riemann Surfaces ◽

Central Charge ◽

Perturbation Expansion ◽

Liouville Theory ◽

Energy Tensor ◽

Ward Identities ◽

Generating Functional ◽

Local Index

We continue the study of quantum Liouville theory through Polyakov’s functional integral,1,2 started in Ref. 3. We derive the perturbation expansion for Schwinger’s generating functional for connected multi-point correlation functions involving stress-energy tensor, give the “dynamical” proof of the Virasoro symmetry of the theory and compute the value of the central charge, confirming previous calculation in Ref. 3. We show that conformal Ward identities for these correlation functions contain such basic facts from Kähler geometry of moduli spaces of Riemann surfaces, as relation between accessory parameters for the Fuchsian uniformization, Liouville action and Eichler integrals, Kähler potential for the Weil-Petersson metric, and local index theorem. These results affirm the fundamental role that universal Ward identities for the generating functional play in Friedan-Shenker modular geometry.4

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Geometric Langlands correspondence for , over the pair of pants

Compositio Mathematica ◽

10.1112/s0010437x18007893 ◽

2019 ◽

Vol 155 (2) ◽

pp. 324-371 ◽

Cited By ~ 1

Author(s):

David Nadler ◽

Zhiwei Yun

Keyword(s):

Projective Line ◽

Langlands Correspondence ◽

Geometric Langlands ◽

Rank One ◽

Tame Ramification

We establish the geometric Langlands correspondence for rank-one groups over the projective line with three points of tame ramification.

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LANGLANDS DUALITY IN LIOUVILLE-$H^3_+$ WZNW CORRESPONDENCE

International Journal of Modern Physics A ◽

10.1142/s0217751x09044607 ◽

2009 ◽

Vol 24 (16n17) ◽

pp. 3137-3170 ◽

Cited By ~ 8

Author(s):

GASTON GIRIBET ◽

YU NAKAYAMA ◽

LORENA NICOLÁS

Keyword(s):

Correlation Functions ◽

Free Field ◽

Liouville Theory ◽

Hamiltonian Reduction ◽

Langlands Correspondence ◽

Conformal Field ◽

Langlands Duality ◽

Dual Version ◽

Free Field Realization ◽

Wznw Model

We show a physical realization of the Langlands duality in correlation functions of [Formula: see text] WZNW model. We derive a dual version of the Stoyanovky–Riabult–Teschner (SRT) formula that relates the correlation function of the [Formula: see text] WZNW and the dual Liouville theory to investigate the level duality k - 2 → (k - 2)-1 in the WZNW correlation functions. Then, we show that such a dual version of the [Formula: see text]-Liouville relation can be interpreted as a particular case of a biparametric family of nonrational conformal field theories (CFT's) based on the Liouville correlation functions, which was recently proposed by Ribault. We study symmetries of these new nonrational CFT's and compute correlation functions explicitly by using the free field realization to see how a generalized Langlands duality manifests itself in this framework. Finally, we suggest an interpretation of the SRT formula as realizing the Drinfeld–Sokolov Hamiltonian reduction. Again, the Hamiltonian reduction reveals the Langlands duality in the [Formula: see text] WZNW model. Our new identity for the correlation functions of [Formula: see text] WZNW model may yield a first step to understand quantum geometric Langlands correspondence yet to be formulated mathematically.

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